These are the same/similar problems from Section 4.2. However, you are asked to solve them with the methods of Section 4.3.
ODE = Ordinary Differential Equation;
IVP = Initial Value Problem
1.
Our first spring-mass system from Section 4.2 has mass \(2\) kg, frictional constant \(10\text{,}\) and spring constant \(12\) kg/s\(^2\text{.}\) It is connected to a motor that provides a force \(f(t) = 20 \sin(t)\) to our system. The spring is stretched to a length of \(1\) m from rest and then released.
Use complexification to find the particular solution to the IVP you found in Section 4.2.
What advantage(s) and disadvantage(s) does this method of complexification have over the method of undetermined coefficients that you used in Section 4.2?
Why is the particular solution also called the steady-state solution? Explain in the context of this problem
2.
Our second spring-mass system from Section 4.2 has mass 1 kg, no frictional forces, and a spring constant \(k = 9\) kg/s\(^2\text{.}\) The spring is stretched to a length of \(1\) m from rest and then released. The laboratory in which we are analyzing our spring-mass system is on a research boat, and on the third day the motion of the waves provides a force \(f(t) = \sin(t) + \cos(3t)\) to our system.
Use complexification to find the particular solution to this IVP.
Write your answer from a. in the form of \(y_p=A\cos{(\omega t-\phi)}\)
How can your solution from b. help you to easier graph your particular solution and the phase diagram?
Use your solution from b. to graph your particular solution.
Use your solution from b. to graph the phase diagram of your particular solution.
What does this phase diagram show you in the context of this problem?
